Niko has an outdoor play tent in the form of a regular triangular pyramid, covered in fabric on all four sides. The surface area of the tent is 100 ft^2., The base is 6 ft., and the slant height is 8 ft. What is the height of the base to the nearest tenth?
The formula for the surface area of a triangular pyramid is given by:
A = (1/2) * P * l + B
where A is the surface area, B is the area of the base, P is the perimeter of the base, and l is the slant height.
In this case, we are given A = 100 ft^2, B = ?, P = 6 ft, and l = 8 ft.
Substituting the given values into the formula, we have:
100 = (1/2) * 6 * 8 + B
Multiplying, we have:
100 = 24 + B
Subtracting 24 from both sides, we have:
76 = B
Therefore, the area of the base is 76 ft^2.
The area of a triangle is given by:
A = (1/2) * b * h
where A is the area, b is the base, and h is the height.
In this case, we are given A = 76 ft^2 and b = 6 ft.
Substituting the given values into the formula, we have:
76 = (1/2) * 6 * h
Multiplying, we have:
76 = 3h
Dividing both sides by 3, we have:
h = 25.333...
Rounding to the nearest tenth, we have:
h ≈ 25.3 ft.
Therefore, the height of the base is approximately 25.3 feet.